Respuesta :
Answer: (-2, -16)
Step-by-step explanation:
1. Y-intercept:
The y-intercept is the point where the graph of the quadratic function crosses the y-axis. To find the y-intercept, we set x = 0 and solve for y.
When x = 0:
f(0) = (0)^2 + 4(0) - 12
f(0) = -12
So, the y-intercept is at the point (0, -12).
2. X-intercepts:
The x-intercepts, also known as the roots or zeros of the quadratic, are the points where the graph intersects the x-axis. To find the x-intercepts, we set y = 0 and solve for x.
When f(x) = 0:
x^2 + 4x - 12 = 0
We can factor this quadratic equation or use the quadratic formula to solve for x. Factoring may not be straightforward, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the given equation, a = 1, b = 4, and c = -12.
x = (-4 ± √(4^2 - 4(1)(-12))) / (2(1))
x = (-4 ± √(16 + 48)) / 2
x = (-4 ± √(64)) / 2
x = (-4 ± 8) / 2
Simplifying further, we have two x-intercepts:
x = (-4 + 8) / 2 = 4 / 2 = 2
x = (-4 - 8) / 2 = -12 / 2 = -6
So, the x-intercepts are at the points (2, 0) and (-6, 0).
3. Vertex:
The vertex is the point on the graph where the quadratic function reaches its maximum or minimum value. The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
For the given equation, a = 1 and b = 4.
x = -4 / (2(1))
x = -4 / 2
x = -2
To find the corresponding y-coordinate of the vertex, we substitute the x-value into the original equation:
f(x) = x^2 + 4x - 12
f(-2) = (-2)^2 + 4(-2) - 12
f(-2) = 4 - 8 - 12
f(-2) = -16
So, the vertex is at the point (-2, -16).
In summary, the key features of the quadratic function f(x) = x^2 + 4x - 12 are:
- Y-intercept: (0, -12)
- X-intercepts: (2, 0) and (-6, 0)
- Vertex: (-2, -16)
